Step | Hyp | Ref
| Expression |
1 | | simpr 109 |
. . . 4
⊢ ((𝜑 ∧ 𝜓) → 𝜓) |
2 | | fprod2d.7 |
. . . 4
⊢ (𝜓 ↔ ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
3 | 1, 2 | sylib 121 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷) |
4 | | nfcv 2299 |
. . . . . 6
⊢
Ⅎ𝑚∏𝑘 ∈ 𝐵 𝐶 |
5 | | nfcsb1v 3064 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐵 |
6 | | nfcsb1v 3064 |
. . . . . . 7
⊢
Ⅎ𝑗⦋𝑚 / 𝑗⦌𝐶 |
7 | 5, 6 | nfcprod 11463 |
. . . . . 6
⊢
Ⅎ𝑗∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
8 | | csbeq1a 3040 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → 𝐵 = ⦋𝑚 / 𝑗⦌𝐵) |
9 | | csbeq1a 3040 |
. . . . . . . 8
⊢ (𝑗 = 𝑚 → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
10 | 9 | adantr 274 |
. . . . . . 7
⊢ ((𝑗 = 𝑚 ∧ 𝑘 ∈ 𝐵) → 𝐶 = ⦋𝑚 / 𝑗⦌𝐶) |
11 | 8, 10 | prodeq12dv 11477 |
. . . . . 6
⊢ (𝑗 = 𝑚 → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶) |
12 | 4, 7, 11 | cbvprodi 11468 |
. . . . 5
⊢
∏𝑗 ∈
{𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 |
13 | | fprod2d.6 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ⊆ 𝐴) |
14 | 13 | unssbd 3286 |
. . . . . . . 8
⊢ (𝜑 → {𝑦} ⊆ 𝐴) |
15 | | vex 2715 |
. . . . . . . . 9
⊢ 𝑦 ∈ V |
16 | 15 | snss 3687 |
. . . . . . . 8
⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴) |
17 | 14, 16 | sylibr 133 |
. . . . . . 7
⊢ (𝜑 → 𝑦 ∈ 𝐴) |
18 | | fprod2d.3 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ Fin) |
19 | 18 | ralrimiva 2530 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 𝐵 ∈ Fin) |
20 | | nfcsb1v 3064 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 |
21 | 20 | nfel1 2310 |
. . . . . . . . . 10
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐵 ∈ Fin |
22 | | csbeq1a 3040 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
23 | 22 | eleq1d 2226 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑦 → (𝐵 ∈ Fin ↔ ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
24 | 21, 23 | rspc 2810 |
. . . . . . . . 9
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 𝐵 ∈ Fin → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin)) |
25 | 17, 19, 24 | sylc 62 |
. . . . . . . 8
⊢ (𝜑 → ⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) |
26 | | fprod2d.4 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑗 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵)) → 𝐶 ∈ ℂ) |
27 | 26 | ralrimivva 2539 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
28 | | nfcsb1v 3064 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 |
29 | 28 | nfel1 2310 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
30 | 20, 29 | nfralw 2494 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
31 | | csbeq1a 3040 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
32 | 31 | eleq1d 2226 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑦 → (𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
33 | 22, 32 | raleqbidv 2664 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑦 → (∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ ↔ ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
34 | 30, 33 | rspc 2810 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (∀𝑗 ∈ 𝐴 ∀𝑘 ∈ 𝐵 𝐶 ∈ ℂ → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
35 | 17, 27, 34 | sylc 62 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
36 | 35 | r19.21bi 2545 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
37 | 25, 36 | fprodcl 11515 |
. . . . . . 7
⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
38 | | csbeq1 3034 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐵 = ⦋𝑦 / 𝑗⦌𝐵) |
39 | | csbeq1 3034 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑦 → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
40 | 39 | adantr 274 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑦 ∧ 𝑘 ∈ ⦋𝑚 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑗⦌𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
41 | 38, 40 | prodeq12dv 11477 |
. . . . . . . 8
⊢ (𝑚 = 𝑦 → ∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
42 | 41 | prodsn 11501 |
. . . . . . 7
⊢ ((𝑦 ∈ 𝐴 ∧ ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
43 | 17, 37, 42 | syl2anc 409 |
. . . . . 6
⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶) |
44 | | nfcv 2299 |
. . . . . . . 8
⊢
Ⅎ𝑚⦋𝑦 / 𝑗⦌𝐶 |
45 | | nfcsb1v 3064 |
. . . . . . . 8
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
46 | | csbeq1a 3040 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → ⦋𝑦 / 𝑗⦌𝐶 = ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
47 | 44, 45, 46 | cbvprodi 11468 |
. . . . . . 7
⊢
∏𝑘 ∈
⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
48 | | csbeq1 3034 |
. . . . . . . . 9
⊢ (𝑚 = (2nd ‘𝑧) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
49 | | snfig 6761 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ V → {𝑦} ∈ Fin) |
50 | 49 | elv 2716 |
. . . . . . . . . 10
⊢ {𝑦} ∈ Fin |
51 | | xpfi 6876 |
. . . . . . . . . 10
⊢ (({𝑦} ∈ Fin ∧
⦋𝑦 / 𝑗⦌𝐵 ∈ Fin) → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
52 | 50, 25, 51 | sylancr 411 |
. . . . . . . . 9
⊢ (𝜑 → ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ∈ Fin) |
53 | | 2ndconst 6171 |
. . . . . . . . . 10
⊢ (𝑦 ∈ 𝐴 → (2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵) |
54 | 17, 53 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → (2nd ↾
({𝑦} ×
⦋𝑦 / 𝑗⦌𝐵)):({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)–1-1-onto→⦋𝑦 / 𝑗⦌𝐵) |
55 | | fvres 5494 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧)) |
56 | 55 | adantl 275 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((2nd ↾ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))‘𝑧) = (2nd ‘𝑧)) |
57 | 45 | nfel1 2310 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ |
58 | 46 | eleq1d 2226 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑚 → (⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ ↔ ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
59 | 57, 58 | rspc 2810 |
. . . . . . . . . 10
⊢ (𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵 → (∀𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ)) |
60 | 35, 59 | mpan9 279 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ⦋𝑦 / 𝑗⦌𝐵) → ⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 ∈ ℂ) |
61 | 48, 52, 54, 56, 60 | fprodf1o 11496 |
. . . . . . . 8
⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
62 | | elxp 4605 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
63 | | nfv 1508 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑧 = 〈𝑚, 𝑘〉 |
64 | | nfv 1508 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑚 ∈ {𝑦} |
65 | 20 | nfcri 2293 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵 |
66 | 64, 65 | nfan 1545 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) |
67 | 63, 66 | nfan 1545 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
68 | 67 | nfex 1617 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
69 | | nfv 1508 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑚∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) |
70 | | opeq1 3743 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
71 | 70 | eqeq2d 2169 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → (𝑧 = 〈𝑚, 𝑘〉 ↔ 𝑧 = 〈𝑗, 𝑘〉)) |
72 | | eleq1w 2218 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 ∈ {𝑦})) |
73 | | velsn 3578 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ {𝑦} ↔ 𝑗 = 𝑦) |
74 | 72, 73 | bitrdi 195 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = 𝑗 → (𝑚 ∈ {𝑦} ↔ 𝑗 = 𝑦)) |
75 | 74 | anbi1d 461 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵))) |
76 | 22 | eleq2d 2227 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
77 | 76 | pm5.32i 450 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) |
78 | 75, 77 | bitr4di 197 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑗 → ((𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵) ↔ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
79 | 71, 78 | anbi12d 465 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = 𝑗 → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
80 | 79 | exbidv 1805 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑗 → (∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)))) |
81 | 68, 69, 80 | cbvexv1 1732 |
. . . . . . . . . . . 12
⊢
(∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑦} ∧ 𝑘 ∈ ⦋𝑦 / 𝑗⦌𝐵)) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
82 | 62, 81 | bitri 183 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) ↔ ∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵))) |
83 | | nfv 1508 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗𝜑 |
84 | | nfcv 2299 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(2nd ‘𝑧) |
85 | 84, 28 | nfcsbw 3067 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
86 | 85 | nfeq2 2311 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑗 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
87 | | nfv 1508 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘𝜑 |
88 | | nfcsb1v 3064 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
89 | 88 | nfeq2 2311 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘 𝐷 =
⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 |
90 | | fprod2d.1 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝐷 = 𝐶) |
91 | 90 | ad2antlr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = 𝐶) |
92 | 31 | ad2antrl 482 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐶 = ⦋𝑦 / 𝑗⦌𝐶) |
93 | | fveq2 5470 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑧 = 〈𝑗, 𝑘〉 → (2nd ‘𝑧) = (2nd
‘〈𝑗, 𝑘〉)) |
94 | | vex 2715 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑗 ∈ V |
95 | | vex 2715 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑘 ∈ V |
96 | 94, 95 | op2nd 6097 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘〈𝑗, 𝑘〉) = 𝑘 |
97 | 93, 96 | eqtr2di 2207 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 〈𝑗, 𝑘〉 → 𝑘 = (2nd ‘𝑧)) |
98 | 97 | ad2antlr 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝑘 = (2nd ‘𝑧)) |
99 | | csbeq1a 3040 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = (2nd ‘𝑧) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
100 | 98, 99 | syl 14 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → ⦋𝑦 / 𝑗⦌𝐶 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
101 | 91, 92, 100 | 3eqtrd 2194 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 = 〈𝑗, 𝑘〉) ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
102 | 101 | expl 376 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
103 | 87, 89, 102 | exlimd 1577 |
. . . . . . . . . . . 12
⊢ (𝜑 → (∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
104 | 83, 86, 103 | exlimd 1577 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑗∃𝑘(𝑧 = 〈𝑗, 𝑘〉 ∧ (𝑗 = 𝑦 ∧ 𝑘 ∈ 𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
105 | 82, 104 | syl5bi 151 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶)) |
106 | 105 | imp 123 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝐷 = ⦋(2nd
‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
107 | 106 | prodeq2dv 11474 |
. . . . . . . 8
⊢ (𝜑 → ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)⦋(2nd ‘𝑧) / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶) |
108 | 61, 107 | eqtr4d 2193 |
. . . . . . 7
⊢ (𝜑 → ∏𝑚 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑚 / 𝑘⦌⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
109 | 47, 108 | syl5eq 2202 |
. . . . . 6
⊢ (𝜑 → ∏𝑘 ∈ ⦋ 𝑦 / 𝑗⦌𝐵⦋𝑦 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
110 | 43, 109 | eqtrd 2190 |
. . . . 5
⊢ (𝜑 → ∏𝑚 ∈ {𝑦}∏𝑘 ∈ ⦋ 𝑚 / 𝑗⦌𝐵⦋𝑚 / 𝑗⦌𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
111 | 12, 110 | syl5eq 2202 |
. . . 4
⊢ (𝜑 → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
112 | 111 | adantr 274 |
. . 3
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷) |
113 | 3, 112 | oveq12d 5844 |
. 2
⊢ ((𝜑 ∧ 𝜓) → (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶) = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
114 | | fprod2d.5 |
. . . . 5
⊢ (𝜑 → ¬ 𝑦 ∈ 𝑥) |
115 | | disjsn 3623 |
. . . . 5
⊢ ((𝑥 ∩ {𝑦}) = ∅ ↔ ¬ 𝑦 ∈ 𝑥) |
116 | 114, 115 | sylibr 133 |
. . . 4
⊢ (𝜑 → (𝑥 ∩ {𝑦}) = ∅) |
117 | | eqidd 2158 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) = (𝑥 ∪ {𝑦})) |
118 | | fprod2dlemstep.x |
. . . . 5
⊢ (𝜑 → 𝑥 ∈ Fin) |
119 | 15 | a1i 9 |
. . . . 5
⊢ (𝜑 → 𝑦 ∈ V) |
120 | | unsnfi 6865 |
. . . . 5
⊢ ((𝑥 ∈ Fin ∧ 𝑦 ∈ V ∧ ¬ 𝑦 ∈ 𝑥) → (𝑥 ∪ {𝑦}) ∈ Fin) |
121 | 118, 119,
114, 120 | syl3anc 1220 |
. . . 4
⊢ (𝜑 → (𝑥 ∪ {𝑦}) ∈ Fin) |
122 | 13 | sselda 3128 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝑗 ∈ 𝐴) |
123 | 26 | anassrs 398 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
124 | 18, 123 | fprodcl 11515 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
125 | 122, 124 | syldan 280 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ∏𝑘 ∈ 𝐵 𝐶 ∈ ℂ) |
126 | 116, 117,
121, 125 | fprodsplit 11505 |
. . 3
⊢ (𝜑 → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶)) |
127 | 126 | adantr 274 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = (∏𝑗 ∈ 𝑥 ∏𝑘 ∈ 𝐵 𝐶 · ∏𝑗 ∈ {𝑦}∏𝑘 ∈ 𝐵 𝐶)) |
128 | | eliun 3855 |
. . . . . . . . . 10
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ↔ ∃𝑗 ∈ 𝑥 𝑧 ∈ ({𝑗} × 𝐵)) |
129 | | xp1st 6115 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ {𝑗}) |
130 | | elsni 3579 |
. . . . . . . . . . . . . 14
⊢
((1st ‘𝑧) ∈ {𝑗} → (1st ‘𝑧) = 𝑗) |
131 | 129, 130 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) = 𝑗) |
132 | 131 | eleq1d 2226 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ({𝑗} × 𝐵) → ((1st ‘𝑧) ∈ 𝑥 ↔ 𝑗 ∈ 𝑥)) |
133 | 132 | biimparc 297 |
. . . . . . . . . . 11
⊢ ((𝑗 ∈ 𝑥 ∧ 𝑧 ∈ ({𝑗} × 𝐵)) → (1st ‘𝑧) ∈ 𝑥) |
134 | 133 | rexlimiva 2569 |
. . . . . . . . . 10
⊢
(∃𝑗 ∈
𝑥 𝑧 ∈ ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
135 | 128, 134 | sylbi 120 |
. . . . . . . . 9
⊢ (𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) → (1st ‘𝑧) ∈ 𝑥) |
136 | | xp1st 6115 |
. . . . . . . . 9
⊢ (𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) → (1st ‘𝑧) ∈ {𝑦}) |
137 | 135, 136 | anim12i 336 |
. . . . . . . 8
⊢ ((𝑧 ∈ ∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
138 | | elin 3291 |
. . . . . . . 8
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ↔ (𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∧ 𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
139 | | elin 3291 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ ((1st ‘𝑧) ∈ 𝑥 ∧ (1st ‘𝑧) ∈ {𝑦})) |
140 | 137, 138,
139 | 3imtr4i 200 |
. . . . . . 7
⊢ (𝑧 ∈ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → (1st ‘𝑧) ∈ (𝑥 ∩ {𝑦})) |
141 | 116 | eleq2d 2227 |
. . . . . . . 8
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) ↔ (1st ‘𝑧) ∈
∅)) |
142 | | noel 3399 |
. . . . . . . . 9
⊢ ¬
(1st ‘𝑧)
∈ ∅ |
143 | 142 | pm2.21i 636 |
. . . . . . . 8
⊢
((1st ‘𝑧) ∈ ∅ → 𝑧 ∈ ∅) |
144 | 141, 143 | syl6bi 162 |
. . . . . . 7
⊢ (𝜑 → ((1st
‘𝑧) ∈ (𝑥 ∩ {𝑦}) → 𝑧 ∈ ∅)) |
145 | 140, 144 | syl5 32 |
. . . . . 6
⊢ (𝜑 → (𝑧 ∈ (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) → 𝑧 ∈ ∅)) |
146 | 145 | ssrdv 3134 |
. . . . 5
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅) |
147 | | ss0 3435 |
. . . . 5
⊢
((∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) ⊆ ∅ → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
148 | 146, 147 | syl 14 |
. . . 4
⊢ (𝜑 → (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∩ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) = ∅) |
149 | | iunxun 3930 |
. . . . . 6
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) |
150 | | nfcv 2299 |
. . . . . . . . 9
⊢
Ⅎ𝑚({𝑗} × 𝐵) |
151 | | nfcv 2299 |
. . . . . . . . . 10
⊢
Ⅎ𝑗{𝑚} |
152 | 151, 5 | nfxp 4615 |
. . . . . . . . 9
⊢
Ⅎ𝑗({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
153 | | sneq 3572 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑚 → {𝑗} = {𝑚}) |
154 | 153, 8 | xpeq12d 4613 |
. . . . . . . . 9
⊢ (𝑗 = 𝑚 → ({𝑗} × 𝐵) = ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵)) |
155 | 150, 152,
154 | cbviun 3888 |
. . . . . . . 8
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ∪
𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) |
156 | | sneq 3572 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑦 → {𝑚} = {𝑦}) |
157 | 156, 38 | xpeq12d 4613 |
. . . . . . . . 9
⊢ (𝑚 = 𝑦 → ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
158 | 15, 157 | iunxsn 3927 |
. . . . . . . 8
⊢ ∪ 𝑚 ∈ {𝑦} ({𝑚} × ⦋𝑚 / 𝑗⦌𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
159 | 155, 158 | eqtri 2178 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ {𝑦} ({𝑗} × 𝐵) = ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵) |
160 | 159 | uneq2i 3259 |
. . . . . 6
⊢ (∪ 𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ∪
𝑗 ∈ {𝑦} ({𝑗} × 𝐵)) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
161 | 149, 160 | eqtri 2178 |
. . . . 5
⊢ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)) |
162 | 161 | a1i 9 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) = (∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵) ∪ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵))) |
163 | | snfig 6761 |
. . . . . . . 8
⊢ (𝑗 ∈ V → {𝑗} ∈ Fin) |
164 | 163 | elv 2716 |
. . . . . . 7
⊢ {𝑗} ∈ Fin |
165 | 122, 18 | syldan 280 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → 𝐵 ∈ Fin) |
166 | | xpfi 6876 |
. . . . . . 7
⊢ (({𝑗} ∈ Fin ∧ 𝐵 ∈ Fin) → ({𝑗} × 𝐵) ∈ Fin) |
167 | 164, 165,
166 | sylancr 411 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ({𝑗} × 𝐵) ∈ Fin) |
168 | 167 | ralrimiva 2530 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
169 | | disjsnxp 6186 |
. . . . . 6
⊢
Disj 𝑗 ∈
(𝑥 ∪ {𝑦})({𝑗} × 𝐵) |
170 | 169 | a1i 9 |
. . . . 5
⊢ (𝜑 → Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) |
171 | | iunfidisj 6892 |
. . . . 5
⊢ (((𝑥 ∪ {𝑦}) ∈ Fin ∧ ∀𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin ∧ Disj 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
172 | 121, 168,
170, 171 | syl3anc 1220 |
. . . 4
⊢ (𝜑 → ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ∈ Fin) |
173 | | eliun 3855 |
. . . . . 6
⊢ (𝑧 ∈ ∪ 𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) ↔ ∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵)) |
174 | | elxp 4605 |
. . . . . . . 8
⊢ (𝑧 ∈ ({𝑗} × 𝐵) ↔ ∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) |
175 | | simprl 521 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑚, 𝑘〉) |
176 | | simprrl 529 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 ∈ {𝑗}) |
177 | | elsni 3579 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 ∈ {𝑗} → 𝑚 = 𝑗) |
178 | 176, 177 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑚 = 𝑗) |
179 | 178 | opeq1d 3749 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 〈𝑚, 𝑘〉 = 〈𝑗, 𝑘〉) |
180 | 175, 179 | eqtrd 2190 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑧 = 〈𝑗, 𝑘〉) |
181 | 180, 90 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 = 𝐶) |
182 | | simpll 519 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝜑) |
183 | 122 | adantr 274 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑗 ∈ 𝐴) |
184 | | simprrr 530 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝑘 ∈ 𝐵) |
185 | 182, 183,
184, 26 | syl12anc 1218 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐶 ∈ ℂ) |
186 | 181, 185 | eqeltrd 2234 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) ∧ (𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵))) → 𝐷 ∈ ℂ) |
187 | 186 | ex 114 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → ((𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
188 | 187 | exlimdvv 1877 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (∃𝑚∃𝑘(𝑧 = 〈𝑚, 𝑘〉 ∧ (𝑚 ∈ {𝑗} ∧ 𝑘 ∈ 𝐵)) → 𝐷 ∈ ℂ)) |
189 | 174, 188 | syl5bi 151 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑥 ∪ {𝑦})) → (𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
190 | 189 | rexlimdva 2574 |
. . . . . 6
⊢ (𝜑 → (∃𝑗 ∈ (𝑥 ∪ {𝑦})𝑧 ∈ ({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
191 | 173, 190 | syl5bi 151 |
. . . . 5
⊢ (𝜑 → (𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵) → 𝐷 ∈ ℂ)) |
192 | 191 | imp 123 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)) → 𝐷 ∈ ℂ) |
193 | 148, 162,
172, 192 | fprodsplit 11505 |
. . 3
⊢ (𝜑 → ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
194 | 193 | adantr 274 |
. 2
⊢ ((𝜑 ∧ 𝜓) → ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷 = (∏𝑧 ∈ ∪
𝑗 ∈ 𝑥 ({𝑗} × 𝐵)𝐷 · ∏𝑧 ∈ ({𝑦} × ⦋𝑦 / 𝑗⦌𝐵)𝐷)) |
195 | 113, 127,
194 | 3eqtr4d 2200 |
1
⊢ ((𝜑 ∧ 𝜓) → ∏𝑗 ∈ (𝑥 ∪ {𝑦})∏𝑘 ∈ 𝐵 𝐶 = ∏𝑧 ∈ ∪
𝑗 ∈ (𝑥 ∪ {𝑦})({𝑗} × 𝐵)𝐷) |